In part 1, the only specifically slow substitution will be the one anchored to end of string, s/[ ]*$//g. Since the following substitutions all shorten the string you'd get a small benefit from doing this last, so combining with the other responses I'd suggest:

while (<IN_FILE>) {
  s/^\ +//;
  s/:000[AP]M|99991231|Jan 1 1900 12:00:00//g;
  s/\ *-\t-\ */-/g;
  s/\ +$//;
}
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For the second part, you can save small amounts: substitute trailing space on the last field after splitting, instead of on the whole line; precalculate the values for $PROC_FIELDS[$i] - 1; avoid the $i loop index; avoid the block in the loop using the modifier form:

my @PROC_INDEXES = map $_ - 1, @PROC_FIELDS;
while (<IN_FILE>) {
  s/^\ +//;
  s/:000[AP]M|99991231|Jan 1 1900 12:00:00//g;
  s/\ *-\t-\ */-/g;
  my @INPUT_FIELDS = split /-/, $_;
  $INPUT_FIELDS[$#INPUT_FIELDS] =~ s/\ +$//;
  /SUN/ && ($_ = '') for @INPUT_FIELDS[@PROC_INDEXES];
  print OUT_FILE join '-', @INPUT_FIELDS;
}
close IN_FILE;
close OUT_FILE;
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In any case, though, all of these savings are quite minor: I suspect the majority of the time is being taken up with unavoidable I/O.

Hugo